ks-package: ks
In ks: Kernel Smoothing
ks-package R Documentation
ks
Description
Kernel smoothing for data from 1- to 6-dimensions.
Details
There are three main types of functions in this package:
computing kernel estimators - these function names begin with ‘k’
computing bandwidth selectors - these begin with ‘h’ (1-d) or
‘H’ (>1-d)
displaying kernel estimators - these begin with ‘plot’.
The kernel used throughout is the normal (Gaussian) kernel K.
For 1-d data, the bandwidth h is the standard deviation of
the normal kernel, whereas for multivariate data, the bandwidth matrix
\bold{{\rm H}} is the variance matrix.
–For kernel density estimation, kde computes
\hat{f}(\bold{x}) = n^{-1} \sum_{i=1}^n K_{\bold{{\rm H}}} (\bold{x} - \bold{X}_i).
The bandwidth matrix \bold{{\rm H}} is a matrix of smoothing
parameters and its choice is crucial for the performance of kernel
estimators. For display, its plot method calls plot.kde.
–For kernel density estimation, there are several varieties of bandwidth selectors
plug-in hpi (1-d);
Hpi, Hpi.diag (2- to 6-d)
least squares (or unbiased) cross validation (LSCV or UCV) hlscv (1-d);
Hlscv, Hlscv.diag (2- to 6-d)
biased cross validation (BCV)
Hbcv, Hbcv.diag (2- to 6-d)
smoothed cross validation (SCV) hscv (1-d);
Hscv, Hscv.diag (2- to 6-d)
normal scale hns (1-d); Hns (2- to 6-d).
–For kernel density support estimation, the main function is
ksupp which is (the convex hull of)
\{\bold{x}: \hat{f}(\bold{x}) >
\tau\}
for a suitable level \tau. This is closely related to the \tau-level set of
\hat{f}.
–For truncated kernel density estimation, the main function is
kde.truncate
\hat{f} (\bold{x}) \bold{1}\{\bold{x} \in \Omega\} /
\int_{\Omega}\hat{f} (\bold{x}) \, d\bold{x}
for a bounded data support \Omega. The standard density
estimate \hat{f} is truncated and rescaled to give
unit integral over \Omega. Its plot method calls plot.kde.
–For boundary kernel density estimation where the kernel function is
modified explicitly in the boundary region, the main function is
kde.boundary
n^{-1} \sum_{i=1}^n K^*_{\bold{{\rm H}}} (\bold{x} - \bold{X}_i)
for a boundary kernel K^*. Its plot method calls plot.kde.
–For variable kernel density estimation where the bandwidth is not a
constant matrix, the main functions are kde.balloon
\hat{f}_{\rm ball}(\bold{x}) = n^{-1} \sum_{i=1}^n K_{\bold{{\rm H}}(\bold{x})} (\bold{x} - \bold{X}_i)
and
kde.sp
\hat{f}_{\rm SP}(\bold{x}) = n^{-1} \sum_{i=1}^n K_{\bold{{\rm H}}(\bold{X}_i)} (\bold{x} - \bold{X}_i).
For the balloon estimation \hat{f}_{\rm ball} the
bandwidth varies with the estimation point \bold{x}, whereas
for the sample point estimation \hat{f}_{\rm SP}
the bandwidth varies with the data point
\bold{X}_i, i=1,\dots,n.
Their plot methods call plot.kde. The bandwidth
selectors for kde.balloon are based on the normal scale bandwidth
Hns(,deriv.order=2) via the MSE minimal formula, and for
kde.SP on Hns(,deriv.order=4) via the Abramson formula.
–For kernel density derivative estimation, the main function is kdde
{\sf D}^{\otimes r}\hat{f}(\bold{x}) = n^{-1} \sum_{i=1}^n {\sf
D}^{\otimes r}K_{\bold{{\rm H}}} (\bold{x} -
\bold{X}_i).
The bandwidth selectors are a modified subset of those for
kde, i.e. Hlscv, Hns, Hpi, Hscv with deriv.order>0.
Its plot method is plot.kdde for plotting each
partial derivative singly.
–For kernel summary curvature estimation, the main function is
kcurv
\hat{s}(\bold{x})= - \bold{1}\{{\sf D}^2 \hat{f}(\bold{x}) <
0\} \mathrm{abs}(|{\sf D}^2 \hat{f}(\bold{x})|)
where {\sf D}^2
\hat{f}(\bold{x}) is the kernel Hessian matrix estimate.
It has the same structure as a kernel density estimate so its plot
method calls plot.kde.
–For kernel discriminant analysis, the main function is
kda which computes density estimates for each the
groups in the training data, and the discriminant surface.
Its plot method is plot.kda. The wrapper function
hkda, Hkda computes
bandwidths for each group in the training data for kde,
e.g. hpi, Hpi.
–For kernel functional estimation, the main function is
kfe which computes the r-th order integrated density functional
\hat{{\bold \psi}}_r = n^{-2} \sum_{i=1}^n \sum_{j=1}^n {\sf D}^{\otimes r}K_{\bold{{\rm H}}}(\bold{X}_i-\bold{X}_j).
The plug-in selectors are hpi.kfe (1-d), Hpi.kfe (2- to 6-d).
Kernel functional estimates are usually not required to computed
directly by the user, but only within other functions in the package.
–For kernel-based 2-sample testing, the main function is
kde.test which computes the integrated
L_2 distance between the two density estimates as the test
statistic, comprising a linear combination of 0-th order kernel
functional estimates:
\hat{T} = \hat{\psi}_{0,1} + \hat{\psi}_{0,2} - (\hat{\psi}_{0,12} +
\hat{\psi}_{0,21}),
and the corresponding p-value. The \psi are
zero order kernel functional estimates with the subscripts indicating
that 1 = sample 1 only, 2 = sample 2 only, and 12, 21 =
samples 1 and 2. The bandwidth selectors are hpi.kfe,
Hpi.kfe with deriv.order=0.
–For kernel-based local 2-sample testing, the main function is
kde.local.test which computes the squared distance
between the two density estimates as the test
statistic
\hat{U}(\bold{x}) = [\hat{f}_1(\bold{x}) -
\hat{f}_2(\bold{x})]^2
and the corresponding local
p-values. The bandwidth selectors are those used with kde,
e.g. hpi, Hpi.
–For kernel cumulative distribution function estimation, the main
function is kcde
\hat{F}(\bold{x}) = n^{-1} \sum_{i=1}^n
\mathcal{K}_{\bold{{\rm H}}} (\bold{x} - \bold{X}_i)
where \mathcal{K} is the integrated kernel.
The bandwidth selectors are hpi.kcde,
Hpi.kcde. Its plot method is
plot.kcde.
There exist analogous functions for the survival function \hat{\bar{F}}.
–For kernel estimation of a ROC (receiver operating characteristic)
curve to compare two samples from \hat{F}_1,
\hat{F}_2, the main function is kroc
\{\hat{F}_{\hat{Y}_1}(z),
\hat{F}_{\hat{Y}_2}(z)\}
based on the cumulative distribution functions of
\hat{Y}_j = \hat{\bar{F}}_1(\bold{X}_j), j=1,2.
The bandwidth selectors are those used with kcde,
e.g. hpi.kcde, Hpi.kcde for
\hat{F}_{\hat{Y}_j}, \hat{\bar{F}}_1. Its plot method
is plot.kroc.
–For kernel estimation of a copula, the
main function is kcopula
\hat{C}(\bold{z}) = \hat{F}(\hat{F}_1^{-1}(z_1), \dots,
\hat{F}_d^{-1}(z_d))
where \hat{F}_j^{-1}(z_j) is
the z_j-th quantile of of the j-th marginal
distribution \hat{F}_j.
The bandwidth selectors are those used with kcde for
\hat{F}, \hat{F}_j.
Its plot method is plot.kcde.
–For kernel mean shift clustering, the main function is
kms. The mean shift recurrence relation of the candidate
point {\bold x}
{\bold x}_{j+1} = {\bold x}_j + \bold{{\rm H}} {\sf D} \hat{f}({\bold
x}_j)/\hat{f}({\bold x}_j),
where j\geq 0 and {\bold x}_0 = {\bold x},
is iterated until {\bold x} converges to its
local mode in the density estimate \hat{f} by following
the density gradient ascent paths. This mode determines the cluster
label for \bold{x}. The bandwidth selectors are those used with
kdde(,deriv.order=1).
–For kernel density ridge estimation, the main function is
kdr. The kernel density ridge recurrence relation of
the candidate point {\bold x}
{\bold x}_{j+1} = {\bold x}_j + \bold{{\rm U}}_{(d-1)}({\bold
x}_j)\bold{{\rm U}}_{(d-1)}({\bold x}_j)^T \bold{{\rm H}} {\sf D}
\hat{f}({\bold x}_j)/\hat{f}({\bold x}_j),
where j\geq 0, {\bold x}_0 = {\bold x} and \bold{{\rm U}}_{(d-1)} is the 1-dimensional projected
density gradient,
is iterated until {\bold x} converges to the ridge in the
density estimate. The bandwidth selectors are those used with
kdde(,deriv.order=2).
– For kernel feature significance, the main function
kfs. The hypothesis test at a point \bold{x} is
H_0(\bold{x}): \mathsf{H} f(\bold{x}) < 0,
i.e. the density Hessian matrix \mathsf{H} f(\bold{x}) is negative definite.
The test statistic is
W(\bold{x}) = \Vert
\mathbf{S}(\bold{x})^{-1/2} \mathrm{vech} \ \mathsf{H} \hat{f} (\bold{x})\Vert ^2
where {\sf H}\hat{f} is the
Hessian estimate, vech is the vector-half operator, and
\mathbf{S} is an estimate of the null variance.
W(\bold{x}) is
approximately \chi^2 distributed with
d(d+1)/2 degrees of freedom.
If H_0(\bold{x}) is rejected, then \bold{x}
belongs to a significant modal region.
The bandwidth selectors are those used with
kdde(,deriv.order=2). Its plot method is
plot.kfs.
–For deconvolution density estimation, the main function is
kdcde. A weighted kernel density
estimation with the contaminated data {\bold W}_1, \dots, {\bold
W}_n,
\hat{f}_w({\bold x}) = n^{-1} \sum_{i=1}^n
\alpha_i K_{\bold{{\rm H}}}({\bold x} - {\bold W}_i),
is utilised, where the weights \alpha_1, \dots,
\alpha_n are chosen via a
quadratic optimisation involving the error variance and the
regularisation parameter. The bandwidth selectors are those used with
kde.
–Binned kernel estimation is an approximation to the exact kernel
estimation and is available for d=1, 2, 3, 4. This makes
kernel estimators feasible for large samples.
–For an overview of this package with 2-d density estimation, see
vignette("kde").
–For ks \geq 1.11.1, the misc3d and
rgl (3-d plot), oz (Australian map) packages, and for ks \geq 1.14.2, the plot3D (3-d plot) package, have been moved from
Depends to Suggests. This was done to allow ks to be installed
on systems where these latter graphical-based packages can't be
installed. Furthermore, since the future of OpenGL in R is not certain,
plot3D becomes the default for 3D plotting for ks \geq 1.12.0. RGL plots are still supported though these may be deprecated in the
future.
Author(s)
Tarn Duong for most of the package.
M. P. Wand for the binned estimation, univariate plug-in selector and
univariate density derivative estimator code.
J. E. Chacon for the unconstrained pilot functional estimation and fast implementation of derivative-based estimation code.
A. and J. Gramacki for the binned estimation for unconstrained bandwidth
matrices.
References
Bowman, A. & Azzalini, A. (1997) Applied Smoothing Techniques
for Data Analysis. Oxford University Press, Oxford.
Chacon, J.E. & Duong, T. (2018) Multivariate Kernel Smoothing
and Its Applications. Chapman & Hall/CRC, Boca Raton.
Duong, T. (2004) Bandwidth Matrices for Multivariate Kernel Density
Estimation. Ph.D. Thesis, University of Western Australia.
Scott, D.W. (2015) Multivariate Density Estimation: Theory,
Practice, and Visualization (2nd edn). John Wiley & Sons, New York.
Silverman, B. (1986) Density Estimation for Statistics and
Data Analysis. Chapman & Hall/CRC, London.
Simonoff, J. S. (1996) Smoothing Methods in Statistics.
Springer-Verlag, New York.
Wand, M.P. & Jones, M.C. (1995) Kernel Smoothing. Chapman &
Hall/CRC, London.
See Also
feature, sm, KernSmooth
ks documentation built on Sept. 30, 2024, 9:15 a.m.
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| ks-package | R Documentation |
ks
Description
Kernel smoothing for data from 1- to 6-dimensions.
Details
There are three main types of functions in this package:
computing kernel estimators - these function names begin with ‘k’
computing bandwidth selectors - these begin with ‘h’ (1-d) or ‘H’ (>1-d)
displaying kernel estimators - these begin with ‘plot’.
The kernel used throughout is the normal (Gaussian) kernel K.
For 1-d data, the bandwidth h is the standard deviation of
the normal kernel, whereas for multivariate data, the bandwidth matrix
\bold{{\rm H}} is the variance matrix.
–For kernel density estimation, kde computes
\hat{f}(\bold{x}) = n^{-1} \sum_{i=1}^n K_{\bold{{\rm H}}} (\bold{x} - \bold{X}_i).
The bandwidth matrix \bold{{\rm H}} is a matrix of smoothing
parameters and its choice is crucial for the performance of kernel
estimators. For display, its plot method calls plot.kde.
–For kernel density estimation, there are several varieties of bandwidth selectors
plug-in
hpi(1-d);Hpi,Hpi.diag(2- to 6-d)least squares (or unbiased) cross validation (LSCV or UCV)
hlscv(1-d);Hlscv,Hlscv.diag(2- to 6-d)biased cross validation (BCV)
Hbcv,Hbcv.diag(2- to 6-d)smoothed cross validation (SCV)
hscv(1-d);Hscv,Hscv.diag(2- to 6-d)normal scale
hns(1-d);Hns(2- to 6-d).
–For kernel density support estimation, the main function is
ksupp which is (the convex hull of)
\{\bold{x}: \hat{f}(\bold{x}) >
\tau\}
for a suitable level \tau. This is closely related to the \tau-level set of
\hat{f}.
–For truncated kernel density estimation, the main function is
kde.truncate
\hat{f} (\bold{x}) \bold{1}\{\bold{x} \in \Omega\} /
\int_{\Omega}\hat{f} (\bold{x}) \, d\bold{x}
for a bounded data support \Omega. The standard density
estimate \hat{f} is truncated and rescaled to give
unit integral over \Omega. Its plot method calls plot.kde.
–For boundary kernel density estimation where the kernel function is
modified explicitly in the boundary region, the main function is
kde.boundary
n^{-1} \sum_{i=1}^n K^*_{\bold{{\rm H}}} (\bold{x} - \bold{X}_i)
for a boundary kernel K^*. Its plot method calls plot.kde.
–For variable kernel density estimation where the bandwidth is not a
constant matrix, the main functions are kde.balloon
\hat{f}_{\rm ball}(\bold{x}) = n^{-1} \sum_{i=1}^n K_{\bold{{\rm H}}(\bold{x})} (\bold{x} - \bold{X}_i)
and
kde.sp
\hat{f}_{\rm SP}(\bold{x}) = n^{-1} \sum_{i=1}^n K_{\bold{{\rm H}}(\bold{X}_i)} (\bold{x} - \bold{X}_i).
For the balloon estimation \hat{f}_{\rm ball} the
bandwidth varies with the estimation point \bold{x}, whereas
for the sample point estimation \hat{f}_{\rm SP}
the bandwidth varies with the data point
\bold{X}_i, i=1,\dots,n.
Their plot methods call plot.kde. The bandwidth
selectors for kde.balloon are based on the normal scale bandwidth
Hns(,deriv.order=2) via the MSE minimal formula, and for
kde.SP on Hns(,deriv.order=4) via the Abramson formula.
–For kernel density derivative estimation, the main function is kdde
{\sf D}^{\otimes r}\hat{f}(\bold{x}) = n^{-1} \sum_{i=1}^n {\sf
D}^{\otimes r}K_{\bold{{\rm H}}} (\bold{x} -
\bold{X}_i).
The bandwidth selectors are a modified subset of those for
kde, i.e. Hlscv, Hns, Hpi, Hscv with deriv.order>0.
Its plot method is plot.kdde for plotting each
partial derivative singly.
–For kernel summary curvature estimation, the main function is
kcurv
\hat{s}(\bold{x})= - \bold{1}\{{\sf D}^2 \hat{f}(\bold{x}) <
0\} \mathrm{abs}(|{\sf D}^2 \hat{f}(\bold{x})|)
where {\sf D}^2
\hat{f}(\bold{x}) is the kernel Hessian matrix estimate.
It has the same structure as a kernel density estimate so its plot
method calls plot.kde.
–For kernel discriminant analysis, the main function is
kda which computes density estimates for each the
groups in the training data, and the discriminant surface.
Its plot method is plot.kda. The wrapper function
hkda, Hkda computes
bandwidths for each group in the training data for kde,
e.g. hpi, Hpi.
–For kernel functional estimation, the main function is
kfe which computes the r-th order integrated density functional
\hat{{\bold \psi}}_r = n^{-2} \sum_{i=1}^n \sum_{j=1}^n {\sf D}^{\otimes r}K_{\bold{{\rm H}}}(\bold{X}_i-\bold{X}_j).
The plug-in selectors are hpi.kfe (1-d), Hpi.kfe (2- to 6-d).
Kernel functional estimates are usually not required to computed
directly by the user, but only within other functions in the package.
–For kernel-based 2-sample testing, the main function is
kde.test which computes the integrated
L_2 distance between the two density estimates as the test
statistic, comprising a linear combination of 0-th order kernel
functional estimates:
\hat{T} = \hat{\psi}_{0,1} + \hat{\psi}_{0,2} - (\hat{\psi}_{0,12} +
\hat{\psi}_{0,21}),
and the corresponding p-value. The \psi are
zero order kernel functional estimates with the subscripts indicating
that 1 = sample 1 only, 2 = sample 2 only, and 12, 21 =
samples 1 and 2. The bandwidth selectors are hpi.kfe,
Hpi.kfe with deriv.order=0.
–For kernel-based local 2-sample testing, the main function is
kde.local.test which computes the squared distance
between the two density estimates as the test
statistic
\hat{U}(\bold{x}) = [\hat{f}_1(\bold{x}) -
\hat{f}_2(\bold{x})]^2
and the corresponding local
p-values. The bandwidth selectors are those used with kde,
e.g. hpi, Hpi.
–For kernel cumulative distribution function estimation, the main
function is kcde
\hat{F}(\bold{x}) = n^{-1} \sum_{i=1}^n
\mathcal{K}_{\bold{{\rm H}}} (\bold{x} - \bold{X}_i)
where \mathcal{K} is the integrated kernel.
The bandwidth selectors are hpi.kcde,
Hpi.kcde. Its plot method is
plot.kcde.
There exist analogous functions for the survival function \hat{\bar{F}}.
–For kernel estimation of a ROC (receiver operating characteristic)
curve to compare two samples from \hat{F}_1,
\hat{F}_2, the main function is kroc
\{\hat{F}_{\hat{Y}_1}(z),
\hat{F}_{\hat{Y}_2}(z)\}
based on the cumulative distribution functions of
\hat{Y}_j = \hat{\bar{F}}_1(\bold{X}_j), j=1,2.
The bandwidth selectors are those used with kcde,
e.g. hpi.kcde, Hpi.kcde for
\hat{F}_{\hat{Y}_j}, \hat{\bar{F}}_1. Its plot method
is plot.kroc.
–For kernel estimation of a copula, the
main function is kcopula
\hat{C}(\bold{z}) = \hat{F}(\hat{F}_1^{-1}(z_1), \dots,
\hat{F}_d^{-1}(z_d))
where \hat{F}_j^{-1}(z_j) is
the z_j-th quantile of of the j-th marginal
distribution \hat{F}_j.
The bandwidth selectors are those used with kcde for
\hat{F}, \hat{F}_j.
Its plot method is plot.kcde.
–For kernel mean shift clustering, the main function is
kms. The mean shift recurrence relation of the candidate
point {\bold x}
{\bold x}_{j+1} = {\bold x}_j + \bold{{\rm H}} {\sf D} \hat{f}({\bold
x}_j)/\hat{f}({\bold x}_j),
where j\geq 0 and {\bold x}_0 = {\bold x},
is iterated until {\bold x} converges to its
local mode in the density estimate \hat{f} by following
the density gradient ascent paths. This mode determines the cluster
label for \bold{x}. The bandwidth selectors are those used with
kdde(,deriv.order=1).
–For kernel density ridge estimation, the main function is
kdr. The kernel density ridge recurrence relation of
the candidate point {\bold x}
{\bold x}_{j+1} = {\bold x}_j + \bold{{\rm U}}_{(d-1)}({\bold
x}_j)\bold{{\rm U}}_{(d-1)}({\bold x}_j)^T \bold{{\rm H}} {\sf D}
\hat{f}({\bold x}_j)/\hat{f}({\bold x}_j),
where j\geq 0, {\bold x}_0 = {\bold x} and \bold{{\rm U}}_{(d-1)} is the 1-dimensional projected
density gradient,
is iterated until {\bold x} converges to the ridge in the
density estimate. The bandwidth selectors are those used with
kdde(,deriv.order=2).
– For kernel feature significance, the main function
kfs. The hypothesis test at a point \bold{x} is
H_0(\bold{x}): \mathsf{H} f(\bold{x}) < 0,
i.e. the density Hessian matrix \mathsf{H} f(\bold{x}) is negative definite.
The test statistic is
W(\bold{x}) = \Vert
\mathbf{S}(\bold{x})^{-1/2} \mathrm{vech} \ \mathsf{H} \hat{f} (\bold{x})\Vert ^2
where {\sf H}\hat{f} is the
Hessian estimate, vech is the vector-half operator, and
\mathbf{S} is an estimate of the null variance.
W(\bold{x}) is
approximately \chi^2 distributed with
d(d+1)/2 degrees of freedom.
If H_0(\bold{x}) is rejected, then \bold{x}
belongs to a significant modal region.
The bandwidth selectors are those used with
kdde(,deriv.order=2). Its plot method is
plot.kfs.
–For deconvolution density estimation, the main function is
kdcde. A weighted kernel density
estimation with the contaminated data {\bold W}_1, \dots, {\bold
W}_n,
\hat{f}_w({\bold x}) = n^{-1} \sum_{i=1}^n
\alpha_i K_{\bold{{\rm H}}}({\bold x} - {\bold W}_i),
is utilised, where the weights \alpha_1, \dots,
\alpha_n are chosen via a
quadratic optimisation involving the error variance and the
regularisation parameter. The bandwidth selectors are those used with
kde.
–Binned kernel estimation is an approximation to the exact kernel estimation and is available for d=1, 2, 3, 4. This makes kernel estimators feasible for large samples.
–For an overview of this package with 2-d density estimation, see
vignette("kde").
–For ks \geq 1.11.1, the misc3d and
rgl (3-d plot), oz (Australian map) packages, and for ks \geq 1.14.2, the plot3D (3-d plot) package, have been moved from
Depends to Suggests. This was done to allow ks to be installed
on systems where these latter graphical-based packages can't be
installed. Furthermore, since the future of OpenGL in R is not certain,
plot3D becomes the default for 3D plotting for ks \geq 1.12.0. RGL plots are still supported though these may be deprecated in the
future.
Author(s)
Tarn Duong for most of the package. M. P. Wand for the binned estimation, univariate plug-in selector and univariate density derivative estimator code. J. E. Chacon for the unconstrained pilot functional estimation and fast implementation of derivative-based estimation code. A. and J. Gramacki for the binned estimation for unconstrained bandwidth matrices.
References
Bowman, A. & Azzalini, A. (1997) Applied Smoothing Techniques for Data Analysis. Oxford University Press, Oxford.
Chacon, J.E. & Duong, T. (2018) Multivariate Kernel Smoothing and Its Applications. Chapman & Hall/CRC, Boca Raton.
Duong, T. (2004) Bandwidth Matrices for Multivariate Kernel Density Estimation. Ph.D. Thesis, University of Western Australia.
Scott, D.W. (2015) Multivariate Density Estimation: Theory, Practice, and Visualization (2nd edn). John Wiley & Sons, New York.
Silverman, B. (1986) Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC, London.
Simonoff, J. S. (1996) Smoothing Methods in Statistics. Springer-Verlag, New York.
Wand, M.P. & Jones, M.C. (1995) Kernel Smoothing. Chapman & Hall/CRC, London.
See Also
feature, sm, KernSmooth
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